superalgebra

and

supergeometry

# Contents

## Idea

Where a system of quantum mechanics is specified by

a system of supersymmetric quantum mechanics has

• a super Hilbert space $ℋ$;

• an odd operator $D$ in $ℋ$, the supercharge

• such that $D\circ D=H$ is the Hamiltonian.

If we regard the Hamiltonian as the generator of the Poincare Lie algebra in one dimension, then the graded commutator $\left[D,D\right]=2H$ is the supersymmetry extension to the super Poincare Lie algebra.

The data of a system of supersymmetric quantum mechanics may also be formalized in terms of a spectral triple.

## References

### Relation to Morse theory

Supersymmetric quantum mechanics gained attention with the work

which showed that Morse theory may be equivalently interpreted as the study of supersymmetric vacua in supersymmetric quantum mechanics, and which was part of what gained Witten the Fields medal 1990. In this article a certain family of deformations of superparticles on a Riemannian manifold are considered and the supersymmetric ground states are shown to be given by the Morse theory of the deformation function.

Reviews include

• Rohit Jain, Supersymmetric Schrödinger operators with applications to Morse theory (pdf)

• Gábor Pete, section 2 of Morse theory, lecture notes 1999-2001 (pdf)

This deformed supersymmetric quantum mechanics arises as the point-particle limit of the type II superstring regarded as quantum mechanics on the loop space, a relation that is stated more explicitly at the end of

• Edward Witten, Global anomalies in string theory, in W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, pp. 61–99. World Scientific, 1985

The relationon between the 2d SCFT describing the type II superstring and this deformed supersymmetric quantum mechanics on loop space has been further explored in

### Relation to index theory

The relation of the partition function of supersymmetric quantum mechanics to index theory was suggested in unpublished work of Edward Witten and formulated in

• Luis Alvarez-Gaumé, Supersymmetry and the Atiyah-Singer index theorem, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. (Euclid)
• Ezra Getzler, Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys. 92 (1983), 163-178. (pdf)
• D. Quillen, Superconnections and the Chern character Topology 24 (1985), no. 1, 89–95;

• Varghese Mathai, Daniel Quillen, Superconnections, Thom classes, and equivariant differential forms. Topology 25 (1986), no. 1, 85–110;

• Ezra Getzler, A short proof of the Atiyah-Singer index theorem, Topology 25 (1986), 111-117 (pdf)

• D. Quillen, Superconnection character forms and the Cayley transform. Topology 27 (1988), no. 2, 211–238